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To solve this problem, we need to determine the smallest amount that Joshua could have spent on burgers and rolls, given the conditions.

### Problem Analysis
1. Joshua bought the same number of burgers as rolls.
2. Burgers are sold in boxes of 12 for £3 per box.
3. Rolls are sold in packets of 9 for £2.50 per packet.

We need to find the smallest quantity of burgers (and therefore rolls) that Joshua could buy such that:
- He buys equal numbers of burgers and rolls.
- This quantity minimizes his total spending.

### Solution Steps

1. **Find the Least Common Multiple (LCM) of Burger and Roll Quantities:**
   - Since burgers come in boxes of 12, Joshua buys burgers in multiples of 12.
   - Since rolls come in packets of 9, Joshua buys rolls in multiples of 9.
   - To find the minimum number of each item he could buy (while having an equal count of burgers and rolls), we calculate the LCM of 12 and 9.

   \[
   \text{LCM}(12, 9) = 36
   \]

   This means the smallest number of burgers (and rolls) he could buy is 36.

2. **Calculate the Number of Boxes and Packets Needed:**
   - **Burgers:** 36 burgers, bought in boxes of 12.
     \[
     \frac{36}{12} = 3 \text{ boxes}
     \]
   - **Rolls:** 36 rolls, bought in packets of 9.
     \[
     \frac{36}{9} = 4 \text{ packets}
     \]

3. **Calculate the Cost:**
   - **Cost of Burgers:** 3 boxes at £3 per box.
     \[
     3 \times 3 = 9 \text{ pounds}
     \]
   - **Cost of Rolls:** 4 packets at £2.50 per packet.
     \[
     4 \times 2.50 = 10 \text{ pounds}
     \]

   So, the total cost is:
   \[
   9 + 10 = 19 \text{ pounds}
   \]

### Answer
The smallest amount that Joshua could have spent in total on burgers and rolls is **£19**.

Would you like further details on any part of the solution?

### Related Questions
1. What if the rolls were sold in packets of 10 instead of 9?
2. How would the answer change if each box of burgers cost £4 instead of £3?
3. If Joshua wanted to buy at least 100 burgers, what would his total cost be?
4. What if Joshua bought a different number of burgers and rolls, e.g., twice as many rolls as burgers?
5. How does changing the LCM calculation affect real-world purchasing scenarios?

### Tip
When minimizing costs in problems involving quantities, always look for the least common multiple to find the smallest amount of items needed while meeting all conditions.

Joshua bought some burgers and rolls for a party. He bought the same number of burgers as rolls. Burgers are sold in boxes of 12, and rolls are sold in packets of 9. Each box of burgers costs £3, and each packet of rolls costs £2.50. What is the smallest amount that Joshua could have spent in total on burgers and rolls? Give your answer in pounds (£).

Find the minimum amount Joshua could spend on burgers and rolls for a party. The problem involves calculating costs based on the least common multiple (LCM) of burgers sold in boxes of 12 and rolls in packets of 9, with each box and packet having set costs. Suitable for students in Grades 5-7.

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